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Investigation of mathematical models for vibrations of one dimensional environments with considering nonlinear resistance forces

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EN
Abstrakty
EN
In this paper we consider important classes of one dimensional environments, bending stiffness of which can be neglected. It is impossible to apply approximate analytical method of solution of mathematical models of dynamic processes. So justification of existence and uniqueness of solutions, carried out a qualitative their evaluation, based on numerical analysis are considering in this paper. Also the features of dynamic processes of some of examined class of systems are analyzed. Methods of qualitative study of oscillations for restricted and unrestricted bodies under the influence of the resistance forces, described in this paper are based on the general principles of the theory of nonlinear boundary value problems – Galerkin method and the method of monotonicity. Scientific novelty consists in generalization these methods of studying for nonlinear problems at new classes of oscillating systems, justification of solution correctness for specified mathematical models that have practical application in real engineering vibration systems.
Twórcy
autor
  • Department of Higher Mathematics; Lviv Polytechnic National University S. Bandery str., 12, Lviv, 79013, Ukraine
  • Hetman Petro Sahaidachny National Army Academy
autor
  • Department of Higher Mathematics; Lviv Polytechnic National University S. Bandery str., 12, Lviv, 79013, Ukraine
  • Department of Higher Mathematics; Lviv Polytechnic National University S. Bandery str., 12, Lviv, 79013, Ukraine
Bibliografia
  • 1. Gajevski H., Greger K., Zakharias K. 1978. Nichtlineare operator gleichungen und operator differen tialgleichungen. Moscow: Mir, 336. (in Russian).
  • 2. Lions J.L. 2002. Some methods for solving nonlinear boundary value problems. Moscow: Editorial URSS, 587 (in Russian).
  • 3. Pukach P.Ya. 2014. Qualitative methods of the investigation of nonlinear oscillation systems. Lviv: Publishing House of Lviv National Polytechnic University, 286. (in Ukrainian).
  • 4. Sleptsova I.P. 2005. Fragmen – Lindelof principle for some quasi-linear evolution equations of second order. Ukrainian Mathematical Journal, Volume 57, Issue 2, 239–249. (in Ukrainian).
  • 5. Agre K. and Rammaha M. A. 2001. Global solutions to boundary value problems for a nonlinear wave equation in high space dimensions. Differential And Integral Equations, Volume 14, 1315−1331.
  • 6. D'Ancona P. and Manfrin R. 1995. A class of locally solvable semilinear equations of weakly hyperbolic type. Ann. Math. Pura Appl., Volume 168, 355–372.
  • 7. Demeio L. and Lenci S. 2007. Forced nonlinear oscillations of semi-infinite cables and beams resting on a unilateral elastic substrate. Nonlinear Dynamics, Volume 49, 203–215.
  • 8. Demeio L. and Lenci S. 2008. Second-order solutions for the dynamics of a semi-infinite cable on a unilateral substrate. Journal of Sound And Vibrations, Volume 315, 414–432.
  • 9. Ghayesh M.H. 2010. Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation. International Journal of Non-Linear Mechanics, Volume 45, 382–394.
  • 10. Pukach P., Kuzio I. and Sokil M. 2013. Qualitative methods for research of transversal vibrations of semi-infinite cable under the action of nonlinear resistance forces. Econtechmod: an international quarterly journal on economics in technology, new technologies and model ling processes. – Lublin–Rzeszow, Vol. 2, No. 1., 43–48.
  • 11. Lavrenyuk S.P. and Pukach P.Ya. 2007. Mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect to space variables. Ukrainian Mathematical Journal, Volume 59, Issue 11, 1708–1718.
  • 12. Lavrenyuk S.P. and Pukach P.Ya. 2007. Variational hyperbolic inequality in the domains unbounded in spatial variables. International Journal of Evolution Equations, Volume 3, Issue 1, 103–122.
  • 13. Pukach P.Ya. 2014. Qualitative research methods of mathematical model of nonlinear vibrations of conveyor belt. Journal of Mathematical Sciences, Volume 198, Issue 1, 31–38.
  • 14. Pukach P.Ya. and Kuzio I.V. 2013. Nonlinear transverse vibrations of semiinfinite cable with consideration paid to resistance. Scientific Bulletin of National Mining University, Issue 3, 82 – 86. (in Ukrainian).
  • 15. Chen L.Q. 2005. Analysis and control of transverse vibrations of axially moving strings. Appl. Mech. Rev, Volume 58, 91–116.
  • 16. Pukach P., Kuzio I. and Nytrebych Z.M. 2013. Influence of some speed parameters on the dynamics of nonlinear flexural vibrations of a drill kolumn ECONTECHMOD, Volume 2, Issue 4, 61–66.
  • 17. Pukach P.Ya. 2012. On the unboundedness of a solution of the mixed problem for a nonlinear evolution equation at a finite time. Nonlinear Oscillations, Volume 14, Issue 3, 369–378.
  • 18. 18. Pukach P.Ya. 2007. Mixed problem for nonlinear equation of beam vibrations type in unbounded domain. Matematychni Studii, Volume 27, no. 2, 139–148. (in Ukrainian).
  • 19. Pukach P. Ya. 2004. Mixed problem in unbounded domain for weakly nonlinear hyperbolic equation with growing coefficients. Matematychni metody i fizyko-mekhanichni polya, Vol. 47, no. 4, 149–154. (in Ukrainian).
  • 20. Pukach P.Ya., Kuzio I.V., Nytrebych Z.M. and Sokhan P.L. 2013. Nonlinear oscillations of elastic beam including dissipation and the Galerkin method in their investigation. Scientific Bulletin of Lviv Polytechnic National University, Series of Dynamics, Strength and Design of Machines and Devices, Volume 759, 106–111. (in Ukrainian).
  • 21. Salenger G. and Vakakis A.F. 1998. Discretness effects in the forced dynamics of a string on a periodic array of non-linear supports. International Journal of Non-Linear Mechanics, Volume 33, 659–673.
  • 22. Santee D.M. and Goncalves P.B. 2006. Oscillations of a beam on a non-linear elastic foundation under periodic loads. Shock and Vibrations, Volume 13, 273–284.
Uwagi
PL
Opracowane ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-b0cf36af-644f-4010-8887-d2b8365f4845
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